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Number of partitions of n in which odd squares occur with 2 types c,c* and with multiplicity 1. The even squares and parts that are twice the squares they occur with multiplicity 1.
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%I #8 Sep 20 2017 05:25:49

%S 1,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,6,8,10,10,10,10,10,10,12,14,16,18,

%T 18,18,18,18,18,22,26,28,30,30,30,30,30,30,34,38,40,42,42,42,44,48,50,

%U 54,58,60,62,62,62,66,74,78,82,86,88,90,90,90

%N Number of partitions of n in which odd squares occur with 2 types c,c* and with multiplicity 1. The even squares and parts that are twice the squares they occur with multiplicity 1.

%C Convolution of A167700 and A167661. - _Vaclav Kotesovec_, Sep 19 2017

%H Vaclav Kotesovec, <a href="/A104274/b104274.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: product_{k>0}((1+x^(2k-1)^2)/(1-x^(2k-1)^2)).

%F a(n) ~ exp(3 * 2^(-8/3) * Pi^(1/3) * ((4-sqrt(2)) * Zeta(3/2))^(2/3) * n^(1/3)) * ((4-sqrt(2)) * Zeta(3/2))^(1/3) / (2^(7/3) * sqrt(3) * Pi^(1/3) * n^(5/6)). - _Vaclav Kotesovec_, Sep 19 2017

%e E.g. a(10)=6 because we can write it as 91,91*,9*1,9*1*,82,811*.

%p series(product((1+x^((2*k-1)^2))/(1-x^(2*k-1)^2)),k=1..100),x=0,100);

%t nmax = 100; CoefficientList[Series[Product[(1 + x^((2*k-1)^2)) / (1 - x^((2*k-1)^2)), {k, 1, Floor[Sqrt[nmax]/2] + 1}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 19 2017 *)

%Y Cf. A080054, A292563.

%K easy,nonn

%O 0,2

%A _Noureddine Chair_, Feb 27 2005